In the coming weeks/months/whenever-I-feel-like-it, I will be performing simulations to evaluate the performance of multi-winner methods. In order to do this, I will make the assumptions that:
* There is a uniform linear political spectrum. (Other models of voter behavior will be considered later.) * Candidates are uniformly-distributed random variables in. * All votes are sincere. (i.e., a voter at position V votes A>B iff abs(A-V) < abs(B-V)) The last two assumptions, that no strategy is involved in either nominations or voting, is admittedly unrealistic. But, as I see it, in order to know the best strategy to use with a method, you must first know how it would behave without strategy, so that's a useful thing to analyze. The measure used to evaluate each method is the expected mean minimum political distance (http://wiki.electorama.com/wiki/Mean_minimum_political_distance). I shall start with the familiar single-vote plurality method. And, for now, I shall limit it to single-winner elections. **** 1 CANDIDATE **** Not much of an election, but quite simple: the one candidate wins. Results for 1,000,000 simulations: Mean = 0.33346117339223746 St. Dev. = 0.074536492271239416 95% C.I. for mean = (0.33331508455185188, 0.33360726223262305) Simplest fraction in C.I. = 1/3 **** 2 CANDIDATES **** def winner(): cand1 = random.uniform(0, 1) cand2 = random.uniform(0, 1) if abs(cand1 - 0.5) < abs(cand2 - 0.5): return cand1 else: return cand2 Results for 1,000,000 simulations: Mean = 0.29158801343581486 St. Dev. = 0.049207169932997265 95% C.I. for mean = (0.29149156915496505, 0.29168445771666468) Simplest fraction in C.I. = 7/24 **** 3 CANDIDATES **** Results for 10,000 simulations (scaled down because now I'm creating files of linear ballots and running them through an election calculator): Mean = 0.29272619476621009 St. Dev. = 0.04956380959184073 95% C.I. for mean = (0.29175476194884398, 0.2936976275835762) Simplest fraction in C.I. = 12/41 **** 4 CANDIDATES **** Results for 10,000 simulations: Mean = 0.29600265035811468 St. Dev. = 0.049225275160608858 95% C.I. for mean = (0.29503785269367599, 0.29696744802255337) Simplest fraction in C.I. = 8/27 **** 5 CANDIDATES **** Results for 10,000 simulations: Mean = 0.30229572580110786 St. Dev. = 0.051063084679029438 95% C.I. for mean = (0.30129490773200368, 0.30329654387021204) Simplest fraction in C.I. = 10/33 **** 6 CANDIDATES **** Results for 10,000 simulations: Mean = 0.30668385765348882 St. Dev. = 0.052246238355619305 95% C.I. for mean = (0.30565985019844172, 0.30770786510853593) Simplest fraction in C.I. = 4/13 **** 7 CANDIDATES **** Results for 10,000 simulations: Mean = 0.31194122641028588 St. Dev. = 0.054292402651190418 95% C.I. for mean = (0.31087711487198105, 0.3130053379485907) Simplest fraction in C.I. = 5/16 **** 8 CANDIDATES **** Results for 10,000 simulations: Mean = 0.31583421713519955 St. Dev. = 0.054691865511392539 95% C.I. for mean = (0.31476227626870318, 0.31690615800169591) Simplest fraction in C.I. = 6/19 **** 9 CANDIDATES **** Results for 10,000 simulations: Mean = 0.31839339216709406 St. Dev. = 0.05584311495999409 95% C.I. for mean = (0.31729888722603289, 0.31948789710815523) Simplest fraction in C.I. = 7/22 **** 10 CANDIDATES **** Results for 10,000 simulations: Mean = 0.32199377022074216 St. Dev. = 0.056722075258496291 95% C.I. for mean = (0.32088203797439191, 0.32310550246709241) Simplest fraction in C.I. = 9/28 **** 11 CANDIDATES **** Results for 10,000 simulations: Mean = 0.3250827093110063 St. Dev. = 0.058751979685123995 95% C.I. for mean = (0.32393119166897361, 0.326234226953039) Simplest fraction in C.I. = 12/37 **** 12 CANDIDATES **** Results for 10,000 simulations: Mean = 0.3256730946327252 St. Dev. = 0.059259775440600918 95% C.I. for mean = (0.32451162437677011, 0.32683456488868029) Simplest fraction in C.I. = 13/40 **** GENERAL OBSERVATION **** I haven't come up with a formula yet, but I have noticed that the lowest (i.e., best) expected MMPD occurs when there are 2 candidates. As more candidates are added, the expected MMPD gets greater (i.e., worse) and appears to approach a limit of 1/3: An election with an extremely large number of candidates is equivalent to a dictatorship. ---- Election-Methods mailing list - see http://electorama.com/em for list info
